Non-Degenerate Multilinear Singular Multipliers with Fractional Rank

We establish $L^p$ estimates for multilinear multipliers acting on $(n-1)$-tuples of functions on ${\mathbb{R}}^d$. We assume that the multiplier satisfies symbol estimates outside a linear subspace of dimension $m$. The difficulty of proving $L^p$ bounds increases with the rank $\frac{m}{d}$, and our focus is on the fractional rank case $\frac{m}{d}<\frac{n}{2}\le \lceil \frac{m}{d}\rceil$.